A Homotopy Equivalence for Partition Posets Related to Liftings of Sn−1-Modules to Sn

We give a homotopy equivalence to explain an Sn−1-module isomorphism which occurs frequently in the homology of subposets of the partition lattice Πn. The isomorphism in question is necessary for the existence of a lifting to Sn of the Sn−1-module involved. It has also been observed in certain deformations of the free Lie algebra. The topological explanation allows us to generate further examples in subposets of Πn. We show that, in these two contexts, it is often accompanied by an Sn-action on the Sn−1-module occurring in the isomorphism. A well-known example is the lifting of the homology of Πn−1.